How do I get rid of the mesh or faces inside my cube?
It looks like you may have inadvertently added a 'Subdivision Surface' Modifier. Go to the properties page and open the 'Modifiers' tab (it will show a little blue wrench). With your cube selected, click the 'X' at the far right of the subdivision surface modifier.
Or, simply hit tab to get into Object mode, delete your cube with X and add a new one.